As for two 8th notes tied together, that's often done across what's called the "imaginary barline," essentially dividing a 4-beat measure into two 2-beat measures. It's done this way to make the downbeats more obviously visible.

The ties are so you can read the actual beat divisions. It would be really tough to read music where, for example, a quarter note started on the AND of beat 1, and the next note started on the AND of beat 2. So, instead, you put two eighth notes tied together, with the second eighth note making it clear where beat two starts.

2 16ths tied together generally just denotes a "slur" or "legato" playing, where you try not to have as much of a separate attack on the note. For the second example, it simply seems like those three notes (the first through third) should be played legato.

One great reason to tie two notes is to show a note that covers separate measures. For example, if you want a quarter note to play on the 8th count of an 8 | 8 time signature in one measure and hold over through the first note of the next measure you'd need to place an eighth note as the last note of the first measure and an eighth note as the first note of the next measure and tie them across the measure divide. Make sense?